Some remarks on the Smoluchowski-Kramers approximation

According to the Smoluchowski-Kramers approximation, solution q(t)(mu) of the equation mu(q)double over dot(t)(mu)=b(q(t)(mu))-(q)over dot(t)(mu)+sigma(q(t)(mu)) (W)over dot(t), q(0)=q, (q)over dot(0)=p, where (W)over dot(t) is the White noise, converges to the solution of equation (q)over dot(t)=b(q(t))+ sigma(q(t))(W)over dot(t), q(0)=q as mudown arrow0. Many asymptotic problems for the last equation were studied in recent years. We consider relations between asymptotics for the first order equation and the original second order equation. Homogenization, large deviations and stochastic resonance, approximation of Brownian motion W-t by a smooth stochastic process, stationary distributions are considered.